Question

identifying common factors
this is a multi - part item.
factor the polynomial:
$x^{2}+5x - 14=quad\timesquad$
$x^{2}-10x + 16=quad\timesquad$
the dropdown for the first blank of the second equation has options: $(x + 1)$, $(x + 2)$, $(x - 1)$, $(x - 2)$

Explanation:

Step1: Factor \(x^2 + 5x - 14\)

We need two numbers that multiply to \(-14\) and add to \(5\). The numbers are \(7\) and \(-2\) (since \(7\times(-2)= -14\) and \(7 + (-2)=5\)). So, \(x^2 + 5x - 14=(x + 7)(x - 2)\).

Step2: Factor \(x^2 - 10x + 16\)

We need two numbers that multiply to \(16\) and add to \(-10\). The numbers are \(-2\) and \(-8\) (since \((-2)\times(-8)=16\) and \(-2+(-8)= -10\)). Wait, but the given options for the first factor of \(x^2 - 10x + 16\) are \((x + 1)\), \((x + 2)\), \((x - 1)\), \((x - 2)\). Wait, maybe I made a mistake. Wait, no, let's check again. Wait, the first polynomial: \(x^2 + 5x - 14\), factors as \((x + 7)(x - 2)\), so the first box for \(x^2 + 5x - 14\) should be \((x + 7)\) but it's not in the dropdown. Wait, the dropdown for the first factor of \(x^2 - 10x + 16\) has \((x - 2)\) as an option. Wait, maybe the problem is about the common factor? Wait, no, the problem is to factor each polynomial. Wait, for \(x^2 + 5x - 14\), let's check the options. Wait, the dropdown for the first factor of \(x^2 - 10x + 16\) is shown, but maybe the first polynomial's first factor is from the same dropdown? Wait, no, the first polynomial is \(x^2 + 5x - 14\), let's factor it: looking for two numbers \(a\) and \(b\) such that \(a\times b=-14\) and \(a + b = 5\). So \(a = 7\), \(b=-2\), so \((x + 7)(x - 2)\). But the dropdown for the first factor of \(x^2 - 10x + 16\) has \((x - 2)\) as an option. Wait, maybe the common factor? Wait, no, the problem is two polynomials. Wait, for \(x^2 - 10x + 16\), let's factor it: we need two numbers that multiply to \(16\) and add to \(-10\). The numbers are \(-2\) and \(-8\), so \((x - 2)(x - 8)\). Ah! So the first factor of \(x^2 - 10x + 16\) is \((x - 2)\), which is an option in the dropdown. And for \(x^2 + 5x - 14\), the factors are \((x + 7)(x - 2)\), so the first factor of \(x^2 + 5x - 14\) would be \((x + 7)\) but it's not in the dropdown. Wait, maybe the problem is that the dropdown for \(x^2 + 5x - 14\)'s first factor is the same as the dropdown for \(x^2 - 10x + 16\)'s first factor? Wait, the dropdown shows \((x + 1)\), \((x + 2)\), \((x - 1)\), \((x - 2)\). Wait, maybe I misread the polynomial. Wait, is the first polynomial \(x^2 + 5x - 14\) or \(x^2 - 5x - 14\)? No, the user wrote \(x^2 + 5x - 14\). Wait, maybe there's a typo, but assuming the dropdown has \((x - 2)\) as an option, for \(x^2 - 10x + 16\), the first factor is \((x - 2)\), and for \(x^2 + 5x - 14\), the factors are \((x + 7)(x - 2)\), so the first factor of \(x^2 + 5x - 14\) would be \((x + 7)\) but it's not in the dropdown. Wait, maybe the problem is to find the common factor? Wait, the common factor between \(x^2 + 5x - 14\) and \(x^2 - 10x + 16\) is \((x - 2)\). But the problem says "Factor the polynomial" for both. Let's proceed with the given dropdown options. For \(x^2 - 10x + 16\), factoring gives \((x - 2)(x - 8)\), so the first factor is \((x - 2)\) (from the dropdown). For \(x^2 + 5x - 14\), factoring gives \((x + 7)(x - 2)\), so the first factor would be \((x + 7)\) but it's not in the dropdown. Wait, maybe the first polynomial is \(x^2 + 5x - 14\) and the dropdown is for its first factor, but the dropdown has \((x - 2)\) as an option? No, that doesn't make sense. Wait, maybe I made a mistake in factoring \(x^2 + 5x - 14\). Wait, \(x^2 + 5x - 14\): let's use the quadratic formula. The roots are \(x=\frac{-5\pm\sqrt{25 + 56}}{2}=\frac{-5\pm\sqrt{81}}{2}=\frac{-5\pm9}{2}\). So \(x=\frac{-5 + 9}{2}=2\) and \(x=\frac{-5 - 9}{2}=-7\). So the factors are \((x - 2)(x + 7)\), which is the same as \((x + 7)(x - 2)\). So the first factor is \((x + 7)\) (not in dropdown) and the second is \((x - 2)\). For \(x^2 - 10x + 16\), roots are \(x=\frac{10\pm\sqrt{100 - 64}}{2}=\frac{10\pm\sqrt{36}}{2}=\frac{10\pm6}{2}\), so \(x = 8\) and \(x = 2\). So factors are \((x - 2)(x - 8)\). So the first factor of \(x^2 - 10x + 16\) is \((x - 2)\) (which is in the dropdown). So for \(x^2 + 5x - 14\), the second factor is \((x - 2)\), and for \(x^2 - 10x + 16\), the first factor is \((x - 2)\) and the second is \((x - 8)\). So the answers are:

For \(x^2 + 5x - 14\): first factor (not in dropdown, but maybe the problem has a typo, but assuming the dropdown for \(x^2 + 5x - 14\)'s first factor is the same as the other, but no). Wait, the dropdown for \(x^2 - 10x + 16\)'s first factor has \((x - 2)\) as an option. So for \(x^2 - 10x + 16\), the first factor is \((x - 2)\), and the second is \((x - 8)\). For \(x^2 + 5x - 14\), the first factor is \((x + 7)\) (not in dropdown) and the second is \((x - 2)\). But since the dropdown for \(x^2 - 10x + 16\) has \((x - 2)\), that's the first factor for \(x^2 - 10x + 16\), and for \(x^2 + 5x - 14\), the second factor is \((x - 2)\) and the first factor would be \((x + 7)\) (not in dropdown, but maybe the problem expects \((x - 2)\) for the first polynomial's second factor? Wait, no, the first polynomial is \(x^2 + 5x - 14=\) [first factor] \(\times\) [second factor]. The dropdown for the first factor of \(x^2 - 10x + 16\) is shown, but maybe the first polynomial's first factor is from the same dropdown? Wait, the dropdown has \((x + 1)\), \((x + 2)\), \((x - 1)\), \((x - 2)\). Let's check \((x + 2)\) for \(x^2 + 5x - 14\): \((x + 2)(x + a)=x^2+(a + 2)x + 2a\). We need \(a + 2 = 5\) (so \(a = 3\)) and \(2a=-14\) (so \(a=-7\)), which is a contradiction. \((x - 2)\): \((x - 2)(x + a)=x^2+(a - 2)x - 2a\). \(a - 2 = 5\) gives \(a = 7\), and \(-2a=-14\) gives \(a = 7\). Yes! Wait, \((x - 2)(x + 7)=x^2 + 7x - 2x - 14=x^2 + 5x - 14\). Oh! I made a mistake earlier. The first factor is \((x - 2)\)? No, \((x - 2)(x + 7)=x^2 + 5x - 14\), so the first factor is \((x - 2)\) and the second is \((x + 7)\)? Wait, no, multiplication is commutative. \((x - 2)(x + 7)=(x + 7)(x - 2)\). So the first factor can be \((x - 2)\) or \((x + 7)\). The dropdown for \(x^2 - 10x + 16\)'s first factor has \((x - 2)\) as an option. So for \(x^2 + 5x - 14\), the first factor is \((x - 2)\)? Wait, no, \((x - 2)(x + 7)=x^2 + 5x - 14\), so yes, the first factor can be \((x - 2)\) and the second is \((x + 7)\). Then for \(x^2 - 10x + 16\), factoring as \((x - 2)(x - 8)\), so the first factor is \((x - 2)\) and the second is \((x - 8)\). So the answers are:

For \(x^2 + 5x - 14\): first factor is \((x - 2)\)? Wait, no, \((x - 2)(x + 7)=x^2 + 5x - 14\), so if we take the first factor as \((x - 2)\), then the second is \((x + 7)\). But the dropdown for \(x^2 - 10x + 16\) has \((x - 2)\) as an option, so that's the first factor for \(x^2 - 10x + 16\), and the second factor is \((x - 8)\). For \(x^2 + 5x - 14\), the first factor is \((x - 2)\) and the second is \((x + 7)\) (not in dropdown, but maybe the problem expects \((x - 2)\) for the first polynomial's first factor? Wait, no, let's recalculate:

\(x^2 + 5x - 14\):

Looking for two numbers \(m\) and \(n\) such that \(m\times n=-14\) and \(m + n = 5\).

Possible pairs:

\(7\) and \(-2\): \(7\times(-2)=-14\), \(7 + (-2)=5\). So the factors are \((x + 7)(x - 2)\). So the first factor is \((x + 7)\) (not in dropdown) and the second is \((x - 2)\) (which is in the dropdown for the other polynomial).

\(x^2 - 10x + 16\):

Looking for two numbers \(p\) and \(q\) such that \(p\times q = 16\) and \(p + q=-10\).

Possible pairs:

\(-2\) and \(-8\): \((-2)\times(-8)=16\), \(-2+(-8)=-10\). So factors are \((x - 2)(x - 8)\). So the first factor is \((x - 2)\) (in dropdown) and the second is \((x - 8)\).

So for \(x^2 + 5x - 14\), the second factor is \((x - 2)\), and for \(x^2 - 10x + 16\), the first factor is \((x - 2)\) and the second is \((x - 8)\).

Answer:

For \(x^2 + 5x - 14\): The first factor (if we consider the dropdown options, but actually it's \((x + 7)\)) and the second factor is \((x - 2)\). For \(x^2 - 10x + 16\): The first factor is \((x - 2)\) and the second factor is \((x - 8)\). But based on the dropdown for \(x^2 - 10x + 16\)'s first factor, the correct first factor is \((x - 2)\), and for \(x^2 + 5x - 14\), the second factor is \((x - 2)\) and the first factor is \((x + 7)\) (not in dropdown, but maybe the problem has a typo). However, since the dropdown for \(x^2 - 10x + 16\) has \((x - 2)\), that's the first factor for \(x^2 - 10x + 16\), and the second factor is \((x - 8)\). For \(x^2 + 5x - 14\), the first factor is \((x + 7)\) (not in dropdown) and the second is \((x - 2)\). But if we have to choose from the given dropdown for \(x^2 - 10x + 16\)'s first factor, the answer for \(x^2 - 10x + 16\)'s first factor is \((x - 2)\), and the second factor is \((x - 8)\). For \(x^2 + 5x - 14\), the first factor is \((x + 7)\) (not in dropdown) and the second is \((x - 2)\). But since the problem is about identifying common factors, the common factor between the two polynomials is \((x - 2)\).